Root Arrangements of Hyperbolic Criterion of Solvability of Finite Dimensional Leibniz Algebras

Abstract

A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P (n) vanishes nowhere. Denote (i) by xk the roots of P (i) , k = 1, . . . , n - i, i = 0, . . . , n - 1. Then in the absence of any equality of the form (j ) (l) xi = xk (1) one has (i) (j ) (i) (2) xk < xk < xk i < j +j -i (the Rolle theorem). For n 4 (resp. for n 5) not all arrangements without (i) equalities (1) of n(n + 1)/2 real numbers xk and compatible with (2) are real izable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and ( 1) ( 1) ( 3) ( 3) ( 1) ( 1) of their derivatives. For n = 5 and when x1 < x2 < x1 < x2 < x3 < x4 we show that from the 40 arrangements without equalities (1) and compatible with (2) only 16 are realizable by HPLFs (from which 6 by perturbations of hyperbolic polynomials and none by hyperbolic polynomials).